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as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at =, because the function is not differentiable there. This suggests that the idea of a solution be extended to allow for solutions that are not everywhere differentiable, thus motivating the following definition.
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve. The Weierstrass function has been historically served the role of a pathological function, being the first published ...
This function is continuous on the closed interval [−r, r] and differentiable in the open interval (−r, r), but not differentiable at the endpoints −r and r. Since f (−r) = f (r), Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero. The theorem applies even when the function cannot be differentiated ...
A classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. [1] The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions.
A Carathéodory-π solution is a generalized solution to an ordinary differential equation. The concept is due to I. Michael Ross and named in honor of Constantin Carathéodory . [ 1 ] Its practicality was demonstrated in 2008 by Ross et al. [ 2 ] in a laboratory implementation of the concept.
For instance, the differential equation dy / dt = y 2 with initial condition y(0) = 1 has the solution y(t) = 1/(1-t), which is not defined at t = 1. Nevertheless, if f is a differentiable function defined over a compact subset of R n, then the initial value problem has a unique solution defined over the entire R. [6]
The key to the concept of weak solution is that there exist functions u that satisfy equation for any , but such u may not be differentiable and so cannot satisfy equation . An example is u(t, x) = | t − x |, as one may check by splitting the integrals over regions x ≥ t and x ≤ t, where u is smooth, and reversing the above computation ...
In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
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